Theorem 23 — duals iff bases correspond to base complements

Let $M$ and $M'$ be matroids on finite sets of elements and let $\sigma$ be a one-to-one correspondence between their elements. Then $M'$ is a dual of $M$ via $\sigma$ (identity (11.1) for every subset) if and only if bases in one correspond to base complements in the other: for every subset $B$ of $M$,

This turns the rank identity (11.1) into a statement about bases only; it is how Whitney proves Theorem 28, and it shows that a matroid's duals are exactly the copies of its usual dual matroid $M^*$.

Formalization Note Whitney's Theorem 23 reads "$M$ and $M'$ are duals if and only if there is a 1–1 correspondence such that bases in one correspond to base complements in the other." It is stated here for a fixed correspondence $\sigma$, as in Whitney's proof; the existential version follows by quantifying over $\sigma$ on both sides. Since $\sigma$ is a bijection, the displayed equivalence for all $B$ covers both directions of "bases in one correspond to base complements in the other". Both ground sets are the whole finite types.